A110162: Riordan array ((1-x)/(1+x), x/(1+x)^2).

Proved for all n all roots real, in (0, 4)interlacing interlacing (weak)alternating coefficient signs

Recurrence from A110162

\(\left(-3 + n\right)\,P_{n}(x) = \left(6 - 3\,x - 2\,n + n x\right)\,P_{n-1}(x) + \left(3 - n\right)\,P_{n-2}(x)\)

\(P_{1}(x) = 1\)

\(P_{2}(x) = -2 + x\)

\(P_{3}(x) = 2 - 4\,x + x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	-2	1														-1
3	2	-4	1													-1
4	-2	9	-6	1												2
5	2	-16	20	-8	1											-1
6	-2	25	-50	35	-10	1										-1
7	2	-36	105	-112	54	-12	1									2
8	-2	49	-196	294	-210	77	-14	1								-1
9	2	-64	336	-672	660	-352	104	-16	1							-1
10	-2	81	-540	1386	-1782	1287	-546	135	-18	1						2
11	2	-100	825	-2640	4290	-4004	2275	-800	170	-20	1					-1
12	-2	121	-1210	4719	-9438	11011	-8008	3740	-1122	209	-22	1				-1
13	2	-144	1716	-8008	19305	-27456	24752	-14688	5814	-1520	252	-24	1			2
14	-2	169	-2366	13013	-37180	63206	-68952	50388	-25194	8645	-2002	299	-26	1		-1
15	2	-196	3185	-20384	68068	-136136	176358	-155040	94962	-40964	12397	-2576	350	-28	1	-1

b-files: A099837 (row sums), A110162 ?

Technical details

Node id: n-afb1686ffa6592c0 · revision 0 · JSON
Definition source: oeis_catalog:A110162
Fingerprint: f6-d603ab5fdaace463

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept -1)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): no
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

A110162 triangle 66 terms, 11 rows · tabl · catalog b-file?

Row sums ? A099837

Submission text ?

Atlas rows n = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:  1;
   2: -2,   1;
   3:  2,  -4,    1;
   4: -2,   9,   -6,    1;
   5:  2, -16,   20,   -8,     1;
   6: -2,  25,  -50,   35,   -10,    1;
   7:  2, -36,  105, -112,    54,  -12,    1;
   8: -2,  49, -196,  294,  -210,   77,  -14,   1;
   9:  2, -64,  336, -672,   660, -352,  104, -16,   1;
  10: -2,  81, -540, 1386, -1782, 1287, -546, 135, -18, 1;

%F Formula

The row polynomials satisfy (n-3)*P(n,x) = -(2*n-6-(n-3)*x)*P(n-1,x) - (n-3)*P(n-2,x) for n >= 4, with P(1,x) = 1, P(2,x) = -2 + x, P(3,x) = 2 - 4*x + x^2.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := -2 + x;
P[3, x_] := 2 - 4*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 6 - (n - 3)*x)*P[n - 1, x] - (n - 3)*P[n - 2, x])/(n - 3)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = -2 + x; v[3] = 2 - 4*x + x^2; for(n = 4, N, my(i = n); v[i] = (-(2*n - 6 - (n - 3)*x)*v[i - 1] - (n - 3)*v[i - 2])/(n - 3)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

A110162 triangle, offset 0, from P₁

%e Example

Triangle begins:
  0:  1;
  1: -2,   1;
  2:  2,  -4,    1;
  3: -2,   9,   -6,    1;
  4:  2, -16,   20,   -8,     1;
  5: -2,  25,  -50,   35,   -10,    1;
  6:  2, -36,  105, -112,    54,  -12,    1;
  7: -2,  49, -196,  294,  -210,   77,  -14,   1;
  8:  2, -64,  336, -672,   660, -352,  104, -16,   1;
  9: -2,  81, -540, 1386, -1782, 1287, -546, 135, -18, 1;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = -(2*n-4-(n-2)*x)*P(n-1,x) - (n-2)*P(n-2,x) for n >= 3, with P(0,x) = 1, P(1,x) = -2 + x, P(2,x) = 2 - 4*x + x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := -2 + x;
P[2, x_] := 2 - 4*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 4 - (n - 2)*x)*P[n - 1, x] - (n - 2)*P[n - 2, x])/(n - 2)]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = -2 + x; v[3] = 2 - 4*x + x^2; for(n = 3, N, my(i = n + 1); v[i] = (-(2*n - 4 - (n - 2)*x)*v[i - 1] - (n - 2)*v[i - 2])/(n - 2)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, +∞)
by Three-term recurrence criterion, Liu–Wang theorem for three-term recurrences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (strict, previous interlaces next) for every m ≥ 3
by Three-term recurrence criterion, Three-term recurrence criterion
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 3
by Liu–Wang theorem for three-term recurrences
Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (0, 4)
by Three-term recurrence criterion
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n(-x) have one sign
derived from Three-term recurrence criterion

Proof chains ? JSON

Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check

Used in 3

References

[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023